Analysis of Synchronization Errors for Differently Parametrized Limit Cycle Oscillators Subject to Bounded Disturbances

For sets of coupled oscillators, the limit cycles can be computed explicitly only in special cases. The existence, the geometry, as well as the properties of the limit cycles depend on the parameterization of oscillators as well as the coupling functions and strength. While most earlier work, which analyzes effects of synchronization only in the limit and for constant coupling gains, recent preceding work of the authors of this paper has proposed a method to determine upper bounds for the synchronization error of oscillators with diffusive coupling based on over-approximations of the reachable sets of the error dynamics. The present paper investigates how upper bounds of the synchronization errors can be determined for differently parameterized oscillators subject to bounded disturbances and depending on different, possibly time-dependent coupling strengths. These extensions considerably broaden the range of applicability of techniques to compute maximum synchronization errors for networked oscillators. In particular, the technique allows to determine if synchronization is maintained by coupling gains varying over time, while ensuring that the synchronization error never exceeds a defined limit. In addition, the paper proposes a method to synthesize time-varying coupling gains, e.g., motivated by minimizing the coupling effort. The findings are applied to a set of different types of coupled oscillators.